Progress on Mazur's Program B -- a horizontal perspective

Abstract: In this talk, I will discuss recent progress on "Mazur's Program B" --- the problem of classifying all possibilities for the image of Galois for an elliptic curve over $\mathbb{Q}$. I will focus on the horizontal perspective of Mazur's Program B, which strives to classify the composite (non-prime power) images of Galois for an elliptic curve over $\mathbb{Q}$. In particular, I will introduce the notion of an entanglement of division fields, give a group theoretic characterization of an entanglement, and describe two sets of joint work. The first is with Harris Daniels where we classify all infinite families of elliptic curves over $\mathbb{Q}$ which have an "unexplained" entanglement between their $p$ and $q$ division fields where $p,q$ are distinct primes, and the second is with Harris Daniels and Álvaro Lozano-Robledo where we prove several results on elliptic curves (and more generally, principally polarized abelian varieties) over $\mathbb{Q}$ when the entanglement occurs over an abelian extension.

number theory

Audience: researchers in the topic

( slides | video )

Rational Points and Galois Representations

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